Math 126C: Week 1 Review
Note: As I’m sure you’ve all seen in lecture, I make mistakes too! I can’t guarantee that everything
here will be right 100% of the time. If you think there’s an error in something, please e-mail me
and I’ll try to get a correction up. Same goes for any clarification or further questions; if I’ve
made something even more confusing, please e-mail or come to office hours to clear it up! Lastly,
these are not meant to be comprehensive reviews; they’re meant to highlight the main topics and
formulas for the week. Doing homework and extra problems is always the best way to prepare and
fully understand the material!
Section 12.1
The right hand rule:
This will be more important later, but for now familiarize yourself with the correct orientation of
the x, y, z axes in R3 .
The distance formula: Given two points P (x1 , y1 , z1 ) and Q(x2 , y2 , z2 ), the distance between the
points is given by the formula
p
|P Q| = (x1 x2 )2 + (y1 y2 )2 + (z1 z2 )2
The equation of a sphere: The equation of a sphere centered at the point (h, k, l) of radius r is
given by the formula
(x h)2 + (y k)2 + (z l)2 = r2
Section 12.2
Basis properties of vectors: See the tables on p.795 of the text.
The magnitude of a vector: The magnitude or length of a two-dimensional vector a = ha1 , a2 i
is given by the formula
q
|a| = a21 + a22
The magnitude or length of a three-dimensional vector a = ha1 , a2 , a3 i is given by the formula
q
|a| = a21 + a22 + a23
Unit vectors: A unit vector is a vector of length 1
Important application: A unit vector u in the direction of a vector v is given by
1
v
.
|v|
Standard basis vectors: The standard basis vectors in R2 are
i = h1, 0i, j = h0, 1i
In R3 , the standard basis vectors are
i = h1, 0, 0i, j = h0, 1, 0i, k = h0, 0, 1i
Note that these are all unit vectors.
Applications:
Look over pages 797 and 798 for a physics application that will be useful for the homework.
Section 12.3
The dot product (definition): If a = ha1 , a2 , a3 i and b = hb1 , b2 , b3 i, then the dot product of a
and b is the number
a · b = a 1 b 1 + a 2 b 2 + a 2 b3
Properties of the dot product: See the table on page 801. A (very) useful alternate definition
of the dot product is
a · b = |a||b| cos ✓
where ✓ is the angle between a and b.
Orthogonality: Two vecotrs are called orthogonal if the angle between them is ⇡/2. The zero
vector is said to be orthogonal to all other vectors. By the definition of the dot product given above,
this is true if and only if their dot product is zero
Projections: The scalar projection of a vector b onto a is given by
compa b =
a·b
|a|
The vector projection of a vector b onto a is given by
✓
◆
a·b a
a·b
proja b =
=
a
|a| |a|
|a|2
Next you’ll find pages from past exams that had questions about this week’s material. They’re all
completely do-able with what we’ve covered so far. The sooner you start practicing exam questions
the better. If you have questions or want to verify your answer, e-mail me or talk to me after class
or come to office hours.
2
Math 126 — Autumn 2011
3
!
!
3. (12 points) Consider the following figure, in which |P Q| = 9 and |QR| = 11.
•
R
120°
•
P
•
Q
!
!
(a) Compute P Q · QR .
(b) Compute the area of the triangle with vertices P , Q, and R.
(c) Which of the following are true? (Check all that apply.)
!
!
P Q ⇥ QR points into the page
!
!
P Q ⇥ QR points out of the page
!
!
!
!
P Q ⇥ QR = QR ⇥ P Q
!
!
!
!
P Q ⇥ QR is parallel to QR ⇥ P Q
!
PQ
!
!
P Q ⇥ QR is orthogonal to the vector
!
PQ
2. (14 points)
(a) (6 pts) Assume a and b are nonzero three-dimensional vectors that are not parallel and are
not perpendicular.
In each case below, determine if the two vectors are always are orthogonal, always are parallel,
always are neither parallel or perpendicular, or it depends on the vectors (meaning depending
on the vectors it is possible they could be perpendicular or parallel or neither).
Circle one for each (no work is necessary):
i. a ⇥ b and 2b.
orthogonal
parallel
neither
depends
ii. a ⇥ b and b ⇥ a.
orthogonal
parallel
neither
depends
iii. proja (b) and b.
orthogonal
parallel
neither
depends
1
a.
|a|
orthogonal
parallel
neither
depends
a.
orthogonal
parallel
neither
depends
iv. proja (b) and
v. a
b and b
(b) (8 pts) Consider the three points A(1, 3, 4), B(0, 2, 1), C(2, 3, 6).
i. Find the area of the triangle determined by the three points.
ii. For this same triangle, find the angle at the corner B.
(Give in degrees rounded to two places after the decimal).
Math 126F
First Midterm.
April 24, 2014
2
1. (11=2+3+3+3 points) Give an example of each of the following. (No
explanation of answers needed for this problem. Be sure to explain your
answers on other problems!)
(a) A nonzero vector v such that projj v = 0
(b) A vector of length 20 that is parallel to 2i − j − 2k. How many such
vectors are there?
(c) A vector that is perpendicular to both i − k and j + k. How many
such vectors are there?
(d) Two nonzero vectors u and v such that |u · v| = |u||v|.